I agree, those voltages sound low. You could have a shorted cell and/or a bad regulator. A 1/2 decent battery with a full charge should be able to hold over 12.5 volts with a >50 watt headlight across it for at least a minute. A good one for probably 30-60 minutes.
Even float charging on a good battery happens at about 12.5-13 volts and 14-15+ under heavy charging after starting or other heavy discharge.
The fact that the system will go above 13 volts means it is likely not the alternator IMHO.
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